Weak global solvability of the two-phase problem for a class of parabolic systems with strong nonlinearity in the gradient. The case of two spatial variables

Arkhipova, A.

doi:10.1090/spmj/1596

Weak global solvability of the two-phase problem for a class of parabolic systems with strong nonlinearity in the gradient. The case of two spatial variables
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by A. A. Arkhipova

St. Petersburg Math. J. 31 (2020), 273-296

DOI: https://doi.org/10.1090/spmj/1596

Published electronically: February 4, 2020

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Abstract:

A class of quasilinear parabolic systems with nondiagonal principal matrix and strongly nonlinear additional terms is considered. The elliptic operator of the system has a variational structure and is generated by a quadratic functional with a nondiagonal matrix. A plane domain of the spatial variables is divided by a smooth curve in two subdomains and the principal matrix of the system has a “jump” when crossing this curve. The two-phase conditions are given on this curve and the Cauchy–Dirichlet conditions hold at the parabolic boundary of the main parabolic cylinder. The existence of a weak Hölder continuous global solution of the two-phase problem is proved. The problem can be regarded as a construction of the heat flow from a given vector-function to an extremal of the functional.

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